A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids

Research output: Contribution to journalArticle

Abstract

In this article, we develop and analyze a new recovery-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the L2 -norm of the solution is of order P+1, when tensor product polynomials of degree at most p are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be P1 +1/2. We use our derivative recovery result to develop a robust recovery-type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery-type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery-type a posteriori error estimates, at a fixed time, converge to the true errors in the L2 -norm under mesh refinement. The order of convergence is proved to be P+1/t Our theoretical results are valid for piecewise polynomials of degree p ≥ 1 and under the condition that each component, |f’i (u) |, i = 1, 2, of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery-based a posteriori error estimator.

Original languageEnglish (US)
Pages (from-to)1224-1265
Number of pages42
JournalNumerical Methods for Partial Differential Equations
Volume33
Issue number4
DOIs
StatePublished - Jul 2017

Fingerprint

Cartesian Grid
A Posteriori Error Estimators
Hyperbolic Conservation Laws
Discontinuous Galerkin Method
Galerkin methods
Conservation
Recovery
Derivatives
Derivative
Directional derivative
Order of Convergence
Polynomials
Enhanced recovery
Fluxes
Norm
Mesh Refinement
A Posteriori Error Estimates
Error Estimator
Piecewise Polynomials
Approximation

Keywords

  • a posteriori error estimator
  • a priori error estimates
  • derivative recovery
  • discontinuous galerkin method
  • nonlinear hyperbolic conservation laws
  • postprocessing
  • superconvergence

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{ead649ddaf354a0aa31e454121f04c8e,
title = "A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids",
abstract = "In this article, we develop and analyze a new recovery-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the L2 -norm of the solution is of order P+1, when tensor product polynomials of degree at most p are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be P1 +1/2. We use our derivative recovery result to develop a robust recovery-type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery-type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery-type a posteriori error estimates, at a fixed time, converge to the true errors in the L2 -norm under mesh refinement. The order of convergence is proved to be P+1/t Our theoretical results are valid for piecewise polynomials of degree p ≥ 1 and under the condition that each component, |f’i (u) |, i = 1, 2, of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery-based a posteriori error estimator.",
keywords = "a posteriori error estimator, a priori error estimates, derivative recovery, discontinuous galerkin method, nonlinear hyperbolic conservation laws, postprocessing, superconvergence",
author = "Mahboub Baccouch",
year = "2017",
month = "7",
doi = "10.1002/num.22141",
language = "English (US)",
volume = "33",
pages = "1224--1265",
journal = "Numerical Methods for Partial Differential Equations",
issn = "0749-159X",
publisher = "John Wiley and Sons Inc.",
number = "4",

}

TY - JOUR

T1 - A posteriori error estimator based on derivative recovery for the discontinuous Galerkin method for nonlinear hyperbolic conservation laws on Cartesian grids

AU - Baccouch, Mahboub

PY - 2017/7

Y1 - 2017/7

N2 - In this article, we develop and analyze a new recovery-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the L2 -norm of the solution is of order P+1, when tensor product polynomials of degree at most p are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be P1 +1/2. We use our derivative recovery result to develop a robust recovery-type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery-type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery-type a posteriori error estimates, at a fixed time, converge to the true errors in the L2 -norm under mesh refinement. The order of convergence is proved to be P+1/t Our theoretical results are valid for piecewise polynomials of degree p ≥ 1 and under the condition that each component, |f’i (u) |, i = 1, 2, of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery-based a posteriori error estimator.

AB - In this article, we develop and analyze a new recovery-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the L2 -norm of the solution is of order P+1, when tensor product polynomials of degree at most p are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be P1 +1/2. We use our derivative recovery result to develop a robust recovery-type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery-type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery-type a posteriori error estimates, at a fixed time, converge to the true errors in the L2 -norm under mesh refinement. The order of convergence is proved to be P+1/t Our theoretical results are valid for piecewise polynomials of degree p ≥ 1 and under the condition that each component, |f’i (u) |, i = 1, 2, of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to show the effectiveness of our recovery-based a posteriori error estimator.

KW - a posteriori error estimator

KW - a priori error estimates

KW - derivative recovery

KW - discontinuous galerkin method

KW - nonlinear hyperbolic conservation laws

KW - postprocessing

KW - superconvergence

UR - http://www.scopus.com/inward/record.url?scp=85017166569&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017166569&partnerID=8YFLogxK

U2 - 10.1002/num.22141

DO - 10.1002/num.22141

M3 - Article

AN - SCOPUS:85017166569

VL - 33

SP - 1224

EP - 1265

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 4

ER -